My point is: why we (different Aikido schools) could agree (to a point) on taijutsu and aikijo, but not on aikiken?

Actually, I have several, quite different bokken I use regularly and have no issue switching back and forth. It's all about the "feel". A few swings and I know how it moves and exactly where the tip is.

a) I don't see much agreement on empty hand

b) I think that only a few of the Founder's students had any jo experience outside the jo forms codified by Saito Sensei. That's why everyone's jo work is so similar. A very few, like Nishio Sensei had soe Jodo.

But many deshi had sword experience of one form or another. Some worked with Inaba doing his personal version of Kashima Ryu. Some, like Saotome Sensei. Imaizumi Sensei, and Chiba Sensei, just as the examples with whom I am familiar had private classes in various koryu. If you look at the post war deshi, many had substantial sword work in their curriculum and not a one was the same as another. Imaizumi Sensei actually taught a whole set of Itto Ryu forms. Saotome Sensei, Chiba Sensei, and Nishio Sensei each emphasized sword work but each had completely unique exercises. O-Sensei never taught a unified sword curriculum and each of the deshi did training on his own to develop his understanding of what the Founder had shown them.